Bistability without hysteresis in chemical reaction systems: A theoretical analysis of irreversible transitions between multiple steady states

The coexistence between two stable steady states, referred to as bistability, is generally associated with a phenomenon of hysteresis in which a system jumps back and forth between the two branches of stable states for different, critical values of some control parameter, corresponding to two limit points. We focus here on the cases where the transitions between the two branches of stable steady states become irreversible when one of the limit points becomes inaccessible or goes to infinity; we refer to these two cases as irreversible transitions of type 1 or 2, respectively. In order to study in detail the conditions in which such irreversible transitions between multiple steady states occur in chemical systems, we analyze two models based on reversible chemical steps. The first model, due to Schlogl, has long been studied as a simple prototype for bistability. This model is shown to admit irreversible transitions of type 1 as one of the limit points associated with bistability moves into a physically inaccessible region of negative values of a control parameter. A second, original model is proposed, to illustrate the case of irreversible transitions of type 2 in which a limit point goes to infinity. Irreversible transitions of type 1 can also occur in this model, as a function of other control parameters. In both models irreversible transitions take place under nonequilibrium conditions. The analysis indicates what reaction steps need to remain reversible in the models in order to preserve the irreversible transitions.